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Properties

Label 20.0.18110217395...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{16}\cdot 5^{15}\cdot 13^{10}$
Root discriminant $29.03$
Ramified primes $3, 5, 13$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![545, 690, 700, 2010, 2446, 1896, 994, -789, -1065, -2484, -578, -357, 828, 3, 202, -63, 74, -18, 13, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 13*x^18 - 18*x^17 + 74*x^16 - 63*x^15 + 202*x^14 + 3*x^13 + 828*x^12 - 357*x^11 - 578*x^10 - 2484*x^9 - 1065*x^8 - 789*x^7 + 994*x^6 + 1896*x^5 + 2446*x^4 + 2010*x^3 + 700*x^2 + 690*x + 545)
 
gp: K = bnfinit(x^20 - 3*x^19 + 13*x^18 - 18*x^17 + 74*x^16 - 63*x^15 + 202*x^14 + 3*x^13 + 828*x^12 - 357*x^11 - 578*x^10 - 2484*x^9 - 1065*x^8 - 789*x^7 + 994*x^6 + 1896*x^5 + 2446*x^4 + 2010*x^3 + 700*x^2 + 690*x + 545, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 13 x^{18} - 18 x^{17} + 74 x^{16} - 63 x^{15} + 202 x^{14} + 3 x^{13} + 828 x^{12} - 357 x^{11} - 578 x^{10} - 2484 x^{9} - 1065 x^{8} - 789 x^{7} + 994 x^{6} + 1896 x^{5} + 2446 x^{4} + 2010 x^{3} + 700 x^{2} + 690 x + 545 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(181102173953389804962158203125=3^{16}\cdot 5^{15}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.03$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{14} - \frac{1}{10} a^{13} - \frac{1}{10} a^{11} - \frac{1}{10} a^{9} - \frac{1}{2} a^{8} + \frac{2}{5} a^{7} - \frac{1}{10} a^{5} + \frac{2}{5} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{15} - \frac{1}{10} a^{13} - \frac{1}{10} a^{12} - \frac{1}{10} a^{11} - \frac{1}{10} a^{10} + \frac{2}{5} a^{9} - \frac{1}{10} a^{8} + \frac{2}{5} a^{7} - \frac{1}{10} a^{6} + \frac{3}{10} a^{5} + \frac{2}{5} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{440} a^{16} + \frac{1}{44} a^{15} - \frac{3}{220} a^{14} - \frac{51}{440} a^{13} + \frac{13}{55} a^{12} + \frac{13}{55} a^{11} - \frac{21}{440} a^{10} - \frac{181}{440} a^{9} + \frac{41}{110} a^{8} - \frac{1}{40} a^{7} + \frac{83}{440} a^{6} + \frac{169}{440} a^{5} - \frac{1}{11} a^{4} - \frac{3}{11} a^{3} + \frac{5}{11} a^{2} - \frac{43}{88} a + \frac{15}{88}$, $\frac{1}{440} a^{17} - \frac{9}{220} a^{15} + \frac{9}{440} a^{14} + \frac{43}{220} a^{13} + \frac{19}{110} a^{12} - \frac{49}{440} a^{11} - \frac{59}{440} a^{10} - \frac{47}{220} a^{9} - \frac{199}{440} a^{8} - \frac{23}{88} a^{7} + \frac{131}{440} a^{6} - \frac{73}{220} a^{5} + \frac{24}{55} a^{4} + \frac{2}{11} a^{3} + \frac{41}{88} a^{2} + \frac{5}{88} a - \frac{9}{44}$, $\frac{1}{749320} a^{18} - \frac{45}{74932} a^{17} + \frac{401}{749320} a^{16} + \frac{20439}{749320} a^{15} + \frac{751}{28820} a^{14} + \frac{66683}{749320} a^{13} - \frac{175913}{749320} a^{12} - \frac{30177}{749320} a^{11} + \frac{154057}{749320} a^{10} + \frac{74079}{374660} a^{9} + \frac{233611}{749320} a^{8} + \frac{3291}{93665} a^{7} + \frac{334941}{749320} a^{6} - \frac{57601}{749320} a^{5} - \frac{7968}{93665} a^{4} - \frac{5281}{13624} a^{3} - \frac{7889}{149864} a^{2} - \frac{68773}{149864} a - \frac{27151}{149864}$, $\frac{1}{513249990198706799581400} a^{19} + \frac{279722572371518101}{513249990198706799581400} a^{18} - \frac{525146998547869347083}{513249990198706799581400} a^{17} - \frac{25335357296024353937}{102649998039741359916280} a^{16} - \frac{9576108444726597349271}{513249990198706799581400} a^{15} - \frac{5375644046743172729807}{513249990198706799581400} a^{14} - \frac{56629798665292703187231}{513249990198706799581400} a^{13} - \frac{1355903766239960775802}{64156248774838349947675} a^{12} + \frac{14436947459648381625953}{64156248774838349947675} a^{11} + \frac{16220796804996238658397}{256624995099353399790700} a^{10} - \frac{12068325485532322422974}{64156248774838349947675} a^{9} + \frac{151819570680241933353903}{513249990198706799581400} a^{8} - \frac{7898666765148761413646}{64156248774838349947675} a^{7} + \frac{113102881177337306314079}{513249990198706799581400} a^{6} + \frac{2034563122118658642219}{10264999803974135991628} a^{5} + \frac{160004795088356268080661}{513249990198706799581400} a^{4} + \frac{14749710883062107421853}{51324999019870679958140} a^{3} + \frac{268684415155303391147}{649683531897097214660} a^{2} + \frac{2861756385891498419593}{7896153695364719993560} a + \frac{159704849385508816723}{470871550641015412460}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2671069.94942 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.21125.1, 5.1.1711125.1 x5, 10.2.14639743828125.1 x5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 5 sibling: 5.1.1711125.1
Degree 10 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$